4 MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN
PROPOSITION 1.2.
Let K be
a
PRC field, and let V
~An
be
an
absolutely
irreducible affine variety defined over K. Let X be a closed set of real closures of
K, one for each ordering of K. Let
X
1
, ... ,
Xm be disjoint clopen subsets of X
that cover
X.
Let
x
1 ,
...
,xm
be nonsingular points on V such that
Xj
E
V(R)
for every R
E
xj,
for
j = 1, ... , m.
Let
vl, ... ,
Vm
E
KX.
Then there is
x E
V(K)
such that for each 1
::::;
j
::::;
m
(1)
llx Xjll
2
vJ
in R,
for every R
E
Xj.
PROOF. Fix j and put
L
=
K(xj)· Let
R
E
Xj· Then
L
~
R.
AsK is dense
in
R
[P2, Proposition 1.4], there is aj E
Kn
such that
(2)
llxj ajll
2
(ir
in
R.
This aj depends on
R,
but if
R' E
Xj is sufficiently close to
R,
then
(2)
holds
also with
R'
instead of
R.
Indeed, the restriction Xj
~
X ( L)
is continuous, and
(2)
describes a basic open set in
X(L).
Use compactness of
Xj
to partition
Xj
into smaller clopen subsets (and thereby increase m). Associate with each of
them the original point Xj such that
(2)
holds with suitable aj for all
R
E
Xj.
By Remark 1.1(a), for each j there is
Cj
E
Kx
such that
Xj
=
H(cj)·
Suppose
that Vis defined by h(X), ...
Jr(X)
E
K[X1, ...
,Xn]·
These together with
the additional polynomials
define an absolutely irreducible variety
W
~
A
n+2m
of dimension dim
V
+
m.
Indeed, by induction on
m
we may assume that
m
=
1. Let x be the generic
point of
V
over
K,
that is, the image of X in in the integral domain K[V]
=
K[XJ/(h, ... ,
fr)·
Let
u
= (
T )2
llx a1ll
2,
and let Y1 be transcendental over
K(V).
Observe that
u
=1
0, since, by
(2), (
T)
2

llx1  a1ll
2
=1
0. Therefore
fr+1(x,y1,Zl)
=
u Yi
c1Zf is irreducible over
K(V)(y1).
Let Z1 be its root
in the algebraic closure
M
of
K(V)(y1).
Clearly
K[WJ
=
K[X, YI, ZI]/(fl, .. · ,
fn fr+I) ~
K[VJ[yi, ZI]
~ M.
It follows that K[W] is an integral domain, and tr.deg.(W)
=
tr.deg.(V)
+
1.
Thus
W
is absolutely irreducible and dim
W
=
dim
V
+
1.
Let
R
E
X.
With no loss, assume
R
E
XI, and hence
R tj.
X2, ... ,
Xm.
Thus,
ci is positive, and c2, ... ,
Cm
are negative in
R.
Apply
(2)
and Remark 1.1(b)
to complete the XI to a point (xi, y, z) E W(R) with Yj
=1
0 for each 1
::::;
j
::::;
m.
In particular,
8£y+J
(xi,y,z)
=1
0: (xi,y,z) is a nonsingular point on
W.
J
By the PRC property of K there exists a point (x, y, z)
E
W(K). Clearly
x E V(K), and for each j we have llxajll
2
::::;
(~
)
2
in R, for each R E Xj· This
and
(2)
imply (1), by Remark 1.1(c). o